An extension the Liénard-Wiechert retardation equations to include the Thomas precession, part 2

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The first paper in this series is at, and it provides essential background material for the following calculations.

There are several ways of computing the derivatives when the differentiations can be performed in either frame of reference, and it is not yet known if the following calculations are of the correct form. The calculations are exploratory.

Derivation of the retardation equations

The following solutions are for the time derivatives of the potentials. The solutions are easily integrated to obtain the potentials in periodic configurations, and the integrations are performed below for a particle in a circular orbit. The solutions are mostly not useful for non-periodic configurations.

In these solutions a is the acceleration of the particle, j=d a/dt, k = d^2 a/dt^2, and l = d^3 a/dt^3. There are no a^3, j^2, or k^1 terms in the rank=3 solution. There are no a^4, j^3, k^2, or l^1 terms in the rank=4 solution. (The terms would drop out if they were carried.)

In neglecting the symmetric terms, the solutions for the tensors of rank 2 and higher are incomplete. The gravitational solutions are under development at

rank 0    0 steps Static Coulomb solution
rank 1    0 steps v^8 The Liénard-Wiechert retardation equations
rank 2    1 step v^8 a^1. The textbook version of the Thomas precession in 4-potential form

rank 3    2 steps v^3 a^2 j^1
rank 3    2 steps v^4 a^2 j^1
rank 3    2 steps v^5 a^2 j^1
rank 3    5 steps v^5 a^2 j^1    same solution as 2 steps
rank 3    2 steps v^12 a^2 j^1
rank 3    2 steps v^12 a^2 j^1    no HTML code

rank 4    3 steps v^4, a^3, j^2, k^1
rank 4    4 steps v^4, a^3, j^2, k^1    same solution as 3 steps
rank 4    3 steps v^8, a^3, j^2, k^1
rank 4    3 steps v^8, a^3, j^2, k^1    no HTML code

steps.a program listing  with HTML links to library functions
steps.a program listing  without HTML links

The solution for the fifth rank tensor is available here, but the discussion in section III at suggests that the fifth rank tensor does not belong in a four dimensional space. The solution is not recommended until the dimensionality of the problem is better understood. The dimensionality of a problem depends on the number of degrees of freedom that it has, so a dimensionality of more than four would not necessarily imply that there are more than three directions in space.

A alternative method of calculation is shown in this section. With this method the Thomas rotation is cumulative in the frame of reference of the particle, meaning that the coordinates are spinning. The potential solution is divergent. The 4-potential and the coordinates transform in the same way, so the calculation indicates that the coordinates in the frame reference of the particle are rotated but not rotating.

The second derivative is not obtainable by applying the rule for the first derivative twice when the chain rule for differentiation is applicable, which is probably why this result is not in agreement with the conventional interpretation of the rotation.

rank 3 differentiation    3 steps v^4 a^2 j^1

diverge.a program listing  with HTML links to library functions

Plausibility checks for the retardation equations

Several plausibility checks are implemented here. The retardation equations appear to be well behaved, but a thorough evaluation will not be possible until the field equations are obtained. Most of the solutions contain only the Thomas terms. The equations are linear, so the Thomas and LW terms can be evaluated separately.

In the solutions for a single charged particle in a circular orbit, the Thomas terms first appear in the r0^3 solutions. The r0^1 and r0^2 solutions are solutions to the Maxwell equations. The r0^3 solutions are solutions are to the restructured Proca equations that were obtained at

The particle velocity cannot exceed c, but there is no upper bound to the angular velocities. The Thomas precession is sensitive to angular velocities. Time derivatives of a in all orders exist in the solution for a particle in a circular orbit. For this reason, the retardation equation for a tensor of any given rank will always fail if enough powers of r0 are carried in the calculation.

The last calculation integrates the third time derivative of the LW equations three times. The only thing accomplished by the calculation is to drop the static terms, but it provides a good workout for a computer program. The third derivatives of the LW equations are the dominant electrical terms in the solutions of the fourth rank tensor.

rank 1 r0^3    LW and Maxwell equations
rank 2 r0^3    restructured Proca equations, Thomas terms only
rank 3 r0^4    Thomas terms only
rank 4 r0^5    Thomas terms only
rank 4 r0^5    LW terms only

wheel.a program listing  with HTML links to library functions
wheel.a program listing  without HTML links

The analogue of the Taylor theorem for the i th difference

This section derives the equations for converting the i th difference to the i th derivative.

First difference
Second difference
Third difference
Fourth difference

curve.a program listing

The multivariate Taylor theorem

In the multivariate Taylor theorem the sum of the exponents of all the terms is the same. Terms with powers higher than that occur in the calculations. They are harmless but should be dropped in the interest of efficiency, and they must be dropped in the final solution. It is all right to drop powers of selected variables in the final solution, so it is also all right to drop the same powers throughout. When the calculations include dt (or delta t), it is only used for computing the derivatives, so it does not count as one of the variables of the multivariate series expansion.

Gary Osborn
Anaheim, California

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If you find technical or conceptual errors in any of the material at this site then please email me the details of it.

Last update 19 Oct 2014      Revision History